Multiplying apparatus



Dec. 12, 1950 w, s Ew 2,533,408

MULTIPLYING APPARATUS Filed Nov. 4, 1947 2 Sheets-Sheet 2 I L FIG. 9 I FIG. IO 30/ 5 335 I |3 302 F I6. I 2

INVENTOR. WILBUR A. STEWARD ATTORNEYS Patented Dec. 12, 1950 MULTIPLYING APPARATUS Wilbur Allro Steward, Omaha, Nebr. .Application November 4, 1947, Serial No. 783,907

1 Claim. 1 This invention relates to multiplying apparatus and more particularly to such apparatus utilizing the basic principle of Napiers rods.

In the several centuries that have passed since the perfection of the multiplying apparatus known as Napiers rods, many different versions have been proposed, but in all such embodiments 'of which I am aware, the arrangement has been such as to tend to cause errors in the mental adding operations which are required in performance of each problem of multiplication. For

the purpose of establishing uniformity of terminology and simplifying the following discussion and description, the following examples of conventional multiplying operations are set forth.

In the multiplication of the factor 347 by the "factor 543, eith'er factor may be designated as the multiplicand, and the other designated as the "multiplier, and each digit of the multiplier is multiplied by each digit of the multiplicand to produce what may be termed a partial product with respect to each such digit by digit multiplication. The several partial products with respect to the multiplication of the digits of the multiplicand by one digit of the multiplier are, in longhand multiplication, mentally added or combined in proper denominational relation to produce "what may be termed an intermediate product which is set down digit by digit as the operation progresses. The successive intermediate products are set down in ofiset denominational relation in accordance with the denominational order of the multiplier digit from which they were respectively derived, after which the several inter- "mediate products are added to ascertain the final product.

' Thus in such an example of longhand multiplication, the written figures will appear as In reality, however, the production of each of I the intermediate products in such a longhand multiplication involves mental operations which zmay be at least partially represented with respect to the first of such intermediate products,

2 Example B Multiplicand 347 Units digit of Multiplier 3 First partial product 21 I Second partial product 12 Third partial product 9 First intermediate product 1041 The basic principle of Napiers rods is of course concerned with the simplification of the performance of operations such as Example B through elimination of the mentally performed digit by digit multiplications to obtain the successive partial products, and in attaining this result in prior art devices employing this principle, individual rods have been provided for each of the digits zero and 1 to 9, and each rod has carried the several partial products of each significant digit times the digit to which such rod is allocated.

These partial products on the several rods have of course been arranged in predetermined positions or spaces on the rods, and the right and left hand digits of each such partial product have been disposed in a particular relation in the space so that when the rods pertaining to the several digits of a multiplicand were placed in a side by side relation, the partial products relating to any selected multiplier digit would be closely related and could be added so as to determine the related intermediate product. Such rods have in the prior art been arranged in some instances to be disposed vertically in a side by side relation, and in other instances to be disposed horizontally in a side by side relation, but in every instance the disposal of the digits of the partial products on such rods has been such as to complicate the addition thereof in ascertaining the intermediate products. It is therefore the primary object of the present invention to simplify the use of partial product tables particularly in the form of rods of the aforesaid general character, and an object related to the foregoing is ,to enable such rods selected according to the digits of a multiplicand to be so arranged as to enable rapid and accurate summation of the partial products to be effected. A further object is to enable the digits of the partial products that are to be added to produce the digits of any order of an intermediate to be disposed in a vertical column, thereby to make the adding operations conform more nearly to ordinary adding operations.

Other and related objects of the present invention are to simplify the relative positioning of groups of rods pertaining to a selected multiplicand; to afford a holder which not only positions and retains the rods while in use but which also affords readily accessible storage spaces for the rods that are not in use; to afford means for denominationally offsetting the partial products on the adjacent rods according to the orders of the multiplicand to which such rods respectively pertain; and to afford guide or directing means of a simple character to aid in accurately selecting the partial products pertaining to a particular multiplier digit.

Other and further objects of the present invention will be apparent from the followin description and claim and are illustrated in the accompanying drawings which, by way of illustration, show preferred embodiments and the principle thereof and what I now consider to be the best mode in'. which 'I" have contemplated applying that principle. Other embodiments of the invention embodying the same or equivalent principle may be used and structural changes may be made as desired by those skilled in the art without departing fromthe present invention and the purview of the appended claim.

fllnthe drawings:

Fig. 1 is a plan view of a multiplying apparatus embodying 'the'features of the invention;

Fig, '2 is a fragmentary vertical sectional view takenalong the line 22 of Fig. 1;

Figs. Sande are left hand and right hand end views respectively" of' the apparatu shown in Fig. 1;

Figs. 5 and 5A are developed and diagrammatic end views' respectively of one of the partial product rods;

Figs. 6 and 6A are developed and diagrammatic end views of another partial product rod;

Figs. '7 and 7A are-developed and diagrammatic end views of another partial product rod;

Fig. 8 is a plan view of another form of multiplyingapparatus having a large and more convenient rod storage arrangement;

Fig. 9 is a fragmentary sectional view taken along the line 9 of Fig. 8;

Fig. 10is a right'hand side-view of the structure'shown inFig. 8;

Fig. 11 is-a view showing the alternative form of partial product rods;

Fig. 12 is a left hand end view of the structure shown in Fig, 11; and

Fig. 13 is a sectional view taken along the line I3-eI-3 of Fig. 11.

For purposes of disclosure, the invention has been illustrated in Figs. 1 to 4 as embodied in a multiplying apparatus I00'having a carrier IOI adapted not only for storage of a plurality of partial. product rods I02, but also for mounting of seleetedof such rods I02onan accessible upper face thereof for the determination of intermediate products of any selected multiplicand factor. The rods I02 as shownlinFi'gs. 1 to 4 are square in cross section, and the carrier'IDI is adapted for cooperation with rods of this form. The carrier IiII is generally rectangular in outline and is relatively thick so that its'front face I03 is spaced substantially from its back face I00, thereby to affordspace for storage openings I that are extended into the body of the carrier from opposite ends thereof. In this-connection it will be observed that the rods I02 are somewhat longer than thecar-rier" IOI, and'the storageopenings I05 opening through oneend face'of'the carrier IN terminate, as at I05E, just short of the other end. Thus the rods I02 project at one end from the carrier IOI so as to be readily accessible for withdrawal as required. The several storage openings I05 open alternately through opposite ends of the carrier IOI so that alternate groups of rods I02 project from opposite ends of the carrier as will be evident in Fig. 1 of the drawings.

The front face I03 of the carrier IOI serves as a mounting surface for those rods I02 that are selected to represent thevarious digits'of a multiplicand factor, and the arrangement is such that the selected rods I02 are disposed on the face I03 in parallel relation with the rods displaced denominationally endwise with respect to each other in accordance with the order of the multiplicand digits that they respectively represent. Thus, in Fig. l, rods I02 that carry the various partial products of the digits 3, 4 and 7 have been illustrated in positions representing the multiplicand factor 347, and this novel arrangement the several partial produ ts rel ted; to h var ous mu tiplier di its. ar disposed in a relationship which simplifies he a ion of the d gits there to ascer n the interm diat p od ct of the selected multiplioand factor 347 times any selected multiplier d it. Th the 3=r0d ich relate inthis ins ance t theh shest O der of the ul iplicand is dispos d p ra l l to and a ainstan upper abutment ledge U0 that is formed in an upstanding relation as a border alongthllpper edge of the front face I03 of the carrier IOI, and the left hand end of the 3rro d is positionedin abutment with a positionin shoulder H formed in an upstanding relation atthe' left hand end of the face I03.

The shoulder H is the upper one of a-series of such shoulder H, H-I to 111-919 which step down to the right as shown in F 1 so tha thclas shoulder H9 is disposed just above a lower'abutment led e I12 which cxtend a an upstanding border along the lower edge of the face I03. Between the ledges- H0 and H2, space is afforded for mounting of ten partial product rods 12,50 that the.'carrier, .as-.shown. has a capac y which e a les t nrder multiplicand factor. The hei ht of the individual shoulder H t H'*-,'9 is equal to the width of the rods I02, a d the horizontau spacin of the should rs H t :I9 is r latediinder the present invention to the p cin -0r disposition of the di its of the partial pr ducts en he rods I02 as will hereinafter be pointed, ou in the description of the specific-structure of the rods, and at th tim the ha act r andadvantageous relation-of, the index and guide margings associated with the face I03, the'ledges H0 and I I2 and the rods I02 willbe described. With further reference howeverto the physical-strife ture of the carrier IOI, it will be observed that retaining or holding: means are afforded on the carrier which serve to hold any selected number of rods I02 in position on'the face 103. Thus a hook I I5 isadapted to engage the'lower edge of the lowermost rod I02 on the face I03, near the i right hand end of the face I03. and a cord II'B connected to the hook H5 extends" upwardly about a guide pulley III that "is fastened tothe upper right hand corner of the ledge III] by a bracket I173." The cord" IIB then extends along the ledge IIO to a guide pulley lIB that-is'secured by a bracket I I0B to the upper left handcorner of the ledge H0. From the pulley H8 thei'co'rd extends along the left handendof the'carrier IN to a guide pulley II9 mounted on a bracket H913, and the end of the cord is led aboutthe 5, pulley I I9 and to the left and into a bore I20 formed in the left end of the carrier IOI. Within the bore I20 the cord H6 is attached to one end of a spring I2I that has its other end anchored near the right hand end of the bore I20, and thus the cord H6 is yieldingly urged so that the hook IIG holds the selected rods I02 in position. In this connection it will be observed that any tendency of the rods I02 to pivot about the right hand endof the ledge I I is prevented by the engagement of the left ends of the rods against the upper faces of the stepped shoulders H.

While a rod I02 might be used to carry the partial products relating to a single multiplicand digit, the use of rods of squarecross section affords four elongated side faces on each rod which may advantageously be utilized so as to provide the partial products of four different multiplicand digits on each such rod. Thus in Figs. 5, 6 and 7, the developed surfaces of three different classes of rods I02 have been shown whereby the capacity of the apparatus may be rendered relatively large without requiring an unduly large number of rods I02. As shown in Figs. and 5A, a class I rod may'have the partial products of the multiplicand digits 1, 2, 3 and 4 formed on the respective side faces thereof in the relation shown in Fig. 5A, and digital identification of the several partial product tables is afforded by identifying digits 125 disposed on the respective faces of .the rod at the right hand ends of such faces.

In forming the figures of the respective partial products of a digit on a side surface of a rod, such elongated side surface is divided into a plurality of equal areas or spaces each of which may be termed a character space, and certain of which are utilized to receive individual digits of each partial product, whiie certain of the spaces serve as separating and guiding areas disposed between the adjacent digits of adjacent partial products. Thus, a plurality of transverse lines I21 are'formed across each side surface upon which partial products are to be formed, these lines being equally spaced in an amount equal to the lateral dimension of the side surface, thereby to form a series of square spaces or areas. The left hand end space in each such side surface is allocated to the multiplier digit 1, as indicated by the designation MP-'-I in Fig. 5, and the multiplicand digit to which the surface is allocatedis formedtherein so that the base of the figure is adjacent and parallel to the lower side edge of such elongated side surface. The next, space to the right,- and every third space thereafter, constitute separating spaces I29, and guide lines I are formed diagonally across such spaces so as to extend from the upper left corner to the lower right corner, and this slope is coordinated as will hereinafter be described with the direction and slope afforded by the'stepped series of shoulders H to H9.

The successive pairs of spaces afforded from left to right between the separating areas I29 are allocated in succession to the multiplier digits 2 to 9, as indicated by the notations MP-Z to MP-9 in Fig. 5, and the digits of the corresponding partial products are disposed in such pairs of spaces in a manner that is related to the disposition of the multiplicand digit in the most left hand space. Thus the left and right hand spaces of each such pair are allocated respectively to the right and left hand digits of the partial products, and such digits are formed in the spaces thus allocated so that the bases of such di its or figures are disposed adjacent, and

parallel to the lower edge of the elongated side surfaceon which they are formed.

With the partial products thus disposed in a line extended along the side surface of the rod, and with separating character spaces between adjacent digits of adjacent partial products, the arrangement of the rods pertaining to selected multiplicand digits in an endwise offset relation in the manner that is shown in Fig. 1, places the digits that areto be added in a vertical column so that the addition is accomplished in a conventional manner. This of course minimizes errors. As an example, in ascertaining the intermediate product of the multiplier digit 9 times the set-up multiplicand factor 347, the units order of the intermediate product may be read off and set down as 3; the figures 6 and 6 disposed one above the other may be read off and added to obtain 12, the right hand digit 2 of which is set down as the tens digit of the intermediate prodnot; the carry-over digit 1 may then be added to the '7 and 3 digits appearing in the next column to obtain 11, the right hand digit 1 of which is set down in the hundreds order of the intermediate product; and the carry-over digit 1 is then added to the 2 appearing in the next column to the left to ascertain the digit 3 which is set 'down in the thousands order of the intermediate product.

Such selection of the digits that are tobe added in determining the intermediate product pertaining to a selected multiplier digit is facilitated by the guide lines I30, which as shown in Fig. 1, form continuous angular guide lines across the group of rods I02 between the partial products relating to the respective multiplier digits, and such guide lines are extended downwardly beyond the selected or set-up rods 592 by angular lines I30A formed on the face tilt of the carrier. At the upper and lower ends of the guide lines I30A, vertical lines I30B and ISGC are formed across the faces of the respective ledges I I53 and H2. The lines I30, I30A and I393 thus afford angular columns within which the partial products relating to the respective multiplier digits are disposed, and these columns are designated by corresponding digits formed in such columns on the ledges H0 and H2 and on the face I03 at spaced points between such ledges. As an additional aid in selecting the proper partial products, the digits pertaining to alternate partial products on each rod I02 may be so formed as to have a constrasting appearance, and this may be done by use of different colors, or as herein shown, by use of light and dark or heavy and light figures in alternate partial products. Thus as shown in Fig. l, the partial products relating to the odd multiplier digits are in heavy type, while the partial products relating to. the even multiplier digits are in light type.

In the form herein illustrated, the guide lines I30 are arranged to slope downwardly from left to right, and this direction of slope is related to the direction of slope of the stepped shoulders H to 1-1-9, and to the order in which the rods pertaining to the respective orders of the multiplicand are disposed on the face I03 of the carrier. Thus, in the form illustrated the rod I02 pertaining to the highest order of the multiplicand is arranged in the uppermost position, and the rods pertaining to the successively lower orders of the multiplicand are arranged successively below the upper rod. The principles of the present invention may, of course, be applied to a reversal of this arrangement, and in such an inembodiment of the invention is r i stance theslope of the guide lines I 30 and the stepped shoulders H to H--9 would be downwardl'y from right to left; and such shoulders in any instance may be disposed for cooperation with either of the ends of the rods I02.-

In Figs. 8 to 10 of the drawings an alternative I illustrated wherein selection of the rods pertaining to a particular multiplicand is facilitated. In this embodiment, a carrier 20l is provided that is similar to the carrier NH but without the holding meanssuch as the hook H5, and this carrier 20! has provision for storage of a plurality of rods I02 in storage spaces 205. Extended from the lower edge of the carrier 20! is a supplemental carrier 250' that is held in place by side bars 25! and 252 that are secured by screws 253 to the sides of the carrier 250 and the ends of the carrier 20!. In the upper face of the supplemental carrier 250, a plurality of spaced horizontal grooves 255 are formed so as to be adapted to receive individual rods I02. The left hand ends of the grooves 255 are closed by the side bar 25!, while the side bar 252 is of less height "so that the right hand end of the grooves 255 are open. The grooves 255 are twelve in number as illustrated in Figs. 8 to 10, so that three zero rods and rods pertaining to the digits '1 to 9 may be disposed respectively in such grooves for rapid selection. 7

In Figs. 11 to 13 another embodiment of the invention is illustrated wherein the carrier is eliminated and the rods are so formed as to embody positioning means for determining the lon gitudinal ofiset of the selected group of partial product rods. Thus as will be evident in Fig. 11, partial product rods 302 are formed with a partial product table pertaining to one selected multiplicand digit formed onone side surface thereof in the manner explained hereinabove, and at the left hand end of each rod, a downward lateralarm 360' is provided which afiords an abutment against which the left end of another rod may bear. The arm: 360- is so formed that the longitudinal offset of the rods will be equal to one I character space, that is, the same as in the embodiment of Fig. 1. As a guide to aid inselec-. tion of the proper group of partial products, a guide rod 30l is provided which has an arm 360A at its left end in the same relation as the arms. 360,, and guide lines 3303, and identification digits 335 are afiorded on the guide rod 3M for cooperation with the angular guide lines I30 of the rods 302'.

' It will be evident that within the purview of It will also be apparent that 6 absence 8v ciples of Napiers rods to be utilized in such a way as to simplify multiplying operations and minimize errors in addition of the partial products. Thus, while I have illustrated and described the preferred embodiments of my invention, it is to be understood that these are capable of var= iation and modification and I therefore do not wish to be limited to the precise details set forth, but desire to avail myself of such changes and alterations as fall within the purview of the fed lowing claim.

I claim:

In a multiplying apparatus operating according' to the broad principle of Napiers rods, a plu-: rality of elongated rods allocated to the respec' tive significant digits and zero and each having" an elongated side surface and having transverse lines extended across said side surface at equal intervals throughout a substantial portion of the length of said side surface to divide said'side' surface into a series of spaces each having a prede termined dimension longitudinally of said sur-' face, every third one of said spaces constituting a separating space and having an angular guide line extended diagonally there'acros's from one upper corner thereof to the opposite lower corner thereof, and the other spaces constitutmg partial product receiving spaces, all of said other spaces except one being arranged in pairs and being allocated in-succession' to the succes sive digits from 1 to 9, the spaces of suchpair's' constituting left and right hand spaces adapted to respectively receive the left and right digits of a partial product, each such divided side surface having the figures of-gthe partial products of the digit to which the rod is allocated times the several digits from 1 to 9 formed thereon in said pairs of spaces in accordance with the digit to which such spaces are allocated and having said figures disposed in a'line such that the bases of such figures are adjacent and parallel to the lower edge of the side surface on which they are formed, and the right and left hand digits of suchfigures being disposed in the related right and left hand spaces in each instance.

WILBUR ALLRO STEWARD;

REFERENCES CITED The following references are of record in the file of this patent:

UNITED STATES PATENTS Name Date Horvath g Jan. 16, 1900 FOREIGN PATENTS Country Date Great Britain 1877" Austria Dec". 10, 1910 Number Number 

